mathematical analysis of great rhombicosidodecahedron (the largest archimedean solid) (application...
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Mathematical Analysis of Great Rhombicosidodecahedron/The Largest Archimedean Solid
Applications of “HCR’s Theory of Polygon” proposed by Mr H.C. Rajpoot (year-2014) ©All rights reserved
Mr Harish Chandra Rajpoot
M.M.M. University of Technology, Gorakhpur-273010 (UP), India March, 2015
Introduction: A great rhombicosidodecahedron is the largest Archimedean solid which has 30 congruent
square faces, 20 congruent regular hexagonal faces & 12 congruent regular decagonal faces each having equal
edge length. It has 180 edges & 120 vertices lying on a spherical surface
with a certain radius. It is created/generated by expanding a truncated
dodecahedron having 20 equilateral triangular faces & 12 regular
decagonal faces. Thus by the expansion, each of 30 originally truncated
edges changes into a square face, each of 20 triangular faces of the original
solid changes into a regular hexagonal face & 12 regular decagonal faces of
the original solid remain unchanged i.e. decagonal faces are shifted radially.
Thus a solid with 30 squares, 20 hexagonal & 12 decagonal faces, is
obtained which is called great rhombicosidodecahedron which is the
largest Archimedean solid. (See figure 1), thus we have
( ) ( )
( ) ( )
( ) ( )
We would apply HCR’s Theory of Polygon to derive a mathematical relationship between radius of the
spherical surface passing through all 120 vertices & the edge length of a great rhombicosidodecahedron for
calculating its important parameters such as normal distance of each face, surface area, volume etc.
Derivation of outer (circumscribed) radius ( ) of great rhombicosidodecahedron:
Let be the radius of the spherical surface passing through all 120 vertices of a great
rhombicosidodecahedron with edge length & the centre O. Consider a square face ABCD with centre ,
regular hexagonal face EFGHIJ with centre & regular decagonal face KLMNPQRSTU with centre (see the
figure 2 below)
Normal distance ( ) of square face ABCD from the centre O of the great rhombicosidodecahedron is
calculated as follows
In right (figure 2)
√( ) ( ) (
√ *
⇒ √( ) (
√ *
√
( )
Figure 1: A great rhombicosidodecahedron having 30 congruent square faces, 20 congruent regular hexagonal faces & 12 congruent regular decagonal faces each of equal edge length 𝒂
Mathematical Analysis of Great Rhombicosidodecahedron/The Largest Archimedean Solid
Applications of “HCR’s Theory of Polygon” proposed by Mr H.C. Rajpoot (year-2014) ©All rights reserved
We know that the solid angle ( ) subtended by any regular polygon with each side of length at any point
lying at a distance H on the vertical axis passing through the centre of plane is given by “HCR’s Theory of
Polygon” as follows
(
√ )
Hence, by substituting the corresponding values in the above expression, we get the solid angle ( )
subtended by each square face (ABCD) at the centre of the great rhombicosidodecahedron as follows
(
(√
)
√ (√
)
)
(√ √
√
√ ( )
, (√
√
+ (√
)
( )
⇒
(
√ ( )
( )
)
(√
) ( )
Figure 2: A square face ABCD, regular hexagonal face EFGHIJ & regular decagonal face KLMNPQRSTU each of edge length 𝒂
Mathematical Analysis of Great Rhombicosidodecahedron/The Largest Archimedean Solid
Applications of “HCR’s Theory of Polygon” proposed by Mr H.C. Rajpoot (year-2014) ©All rights reserved
Similarly, Normal distance ( ) of regular hexagonal face EFGHIJ from the centre O of the great
rhombicosidodecahedron is calculated as follows
In right (figure 2)
√( ) ( ) ( )
⇒ √( ) ( ) √
( )
Hence, by substituting all the corresponding values, the solid angle ( ) subtended by each regular hexagonal
face (EFGHIJ) at the centre of the great rhombicosidodecahedron is given as follows
(
(√
)
√ (√ )
)
(
√
√ (√ )
)
(√
√
+
⇒ (√
) (√
) ( )
Similarly, Normal distance ( ) of regular decagonal face KLMNPQRSTU from the centre O of the great
rhombicosidodecahedron is calculated as follows
In right (figure 2)
⇒ √( ) ( )
( √ )
⇒ √( ) (
( √ ))
√( )
( √ ) √
( √ )
( )
Hence, by substituting all the corresponding values, the solid angle ( ) subtended by each regular decagonal
face (KLMNPQRSTU) at the centre of great rhombicosidodecahedron is given as follows
(
(√
( √ )
,
√ (√
( √ )
,
)
Mathematical Analysis of Great Rhombicosidodecahedron/The Largest Archimedean Solid
Applications of “HCR’s Theory of Polygon” proposed by Mr H.C. Rajpoot (year-2014) ©All rights reserved
(
√ ( √ ) (
√
)
√ ( √ ) (√ √ )
)
(
√ (
√
) ( √ ) (
√
)
√ ( √ ) ( √ )
)
(
√(
√
)
√
)
(
√
( √
)
)
(
√
( √
)
)
( )
Since a great rhombicosidodecahedron is a closed surface & we know that the total solid angle, subtended by
any closed surface at any point lying inside it, is (Ste-radian) hence the sum of solid angles subtended
by 30 congruent square faces, 20 congruent regular hexagonal faces & 12 congruent regular decagonal faces
at the centre of the great rhombicosidodecahedron must be . Thus we have
[ ] [ ] [ ]
Now, by substituting the values of from eq(II), (IV) & (VI) in the above expression we get
* (√
)+ * (√
)+
[
(
√
( √
)
)
]
⇒
[
(√
) (√
)
(
√
( √
)
)
]
⇒ (√
) (√
)
(
√
( √
)
)
Mathematical Analysis of Great Rhombicosidodecahedron/The Largest Archimedean Solid
Applications of “HCR’s Theory of Polygon” proposed by Mr H.C. Rajpoot (year-2014) ©All rights reserved
⇒ (√
) (√
)
(
√
( √
)
)
⇒
(
√
√ (√
)
√
√ (√
)
)
(
√
( √
)
)
( ( √ √ )
)
⇒ (√
√
√
√
)
(
√
(
√
( √
)
)
)
⇒ ( √ ( )
√ ( )
+
(
√
( √
)
)
( √
( √ )
( ))
⇒ √ ( )
√ ( )
√
( √ )
( )
⇒ √ √ ( ) √ √ ( ) √( √ )( )
⇒ (√ ( ) √( ))
(√( √ )( ))
⇒ ( ) ( ) √ ( )( ) ( √ )( )
⇒ ( √ ) ( √ ) √ ( )( )
⇒ ( ( √ ) ( √ ))
( √ ( )( ))
⇒ ( √ ) √ ( √ ) ( )( )
⇒ ( √ ) ( √ ) ( √ )
⇒ ( √ ) ( √ ) ( √ )
Now, solving the above biquadratic equation for the values of as follows
Mathematical Analysis of Great Rhombicosidodecahedron/The Largest Archimedean Solid
Applications of “HCR’s Theory of Polygon” proposed by Mr H.C. Rajpoot (year-2014) ©All rights reserved
⇒ ( ( √ )) √( ( √ ))
( ( √ )) ( ( √ ))
( √ )
( √ ) √ ( √ ) ( √ )( √ )
( √ )
( √ ) √ ( √ ) ( √ )
( √ ) ( √ ) √ √
( √ )
( ( √ ) √ √ ) ( √ )
( √ ) ( √ ) ( √ ) ( √ ) ( √ )√ ( √ )
( )
( √ ) ( √ ) ( √ )( √ )
( √ ) ( √ )
( √ ) ( √ )
( √ ) ( √ )
1. Taking positive sign, we have
( √ ) ( √ )
√
√
√
√ √
Since, hence, the above value is acceptable.
2. Taking negative sign, we have
( √ ) ( √ )
√
⇒ ( )
Hence, the above value is discarded, now we have
√ √ ⇒
√ √
√ √
Hence, outer (circumscribed) radius ( ) of a great rhombicosidodecahedron with edge length is given as
√ √ ( )
Normal distance ( ) of square faces from the centre of great rhombicosidodecahedron: The
normal distance ( ) of each of 30 congruent square faces from the centre O of a great
rhombicosidodecahedron is given from eq(I) as follows
√
√ (
√ √ )
√
√
√ √
√( √ )
( √ )
Mathematical Analysis of Great Rhombicosidodecahedron/The Largest Archimedean Solid
Applications of “HCR’s Theory of Polygon” proposed by Mr H.C. Rajpoot (year-2014) ©All rights reserved
( √ )
It’s clear that all 30 congruent square faces are at an equal normal distance from the centre of any great
rhombicosidodecahedron.
Solid angle ( ) subtended by each of the square faces at the centre of great
rhombicosidodecahedron: solid angle ( ) subtended by each square face is given from eq(II) as follows
(√
) (√
) ( *
Hence, by substituting the corresponding value of in the above expression, we get
(
√ ( √ √ )
( √ √ )
)
(√ √
( √ ))
(√ √
( √ )) (√
( √ )( √ )
( √ )( √ ))
(√ √
)
(
√ √
) (
√
)
Normal distance ( ) of regular hexagonal faces from the centre of great
rhombicosidodecahedron: The normal distance ( ) of each of 20 congruent regular hexagonal faces from
the centre O of a great rhombicosidodecahedron is given from eq(III) as follows
√ √(
√ √ )
√ √
√ √
√ ( √ )
√ ( √ )
√ ( √ )
It’s clear that all 20 congruent regular hexagonal faces are at an equal normal distance from the centre of
any great rhombicosidodecahedron.
Solid angle ( ) subtended by each of the regular hexagonal faces at the centre of great
rhombicosidodecahedron: solid angle ( ) subtended by each regular hexagonal face is given from eq(IV)
as follows
(√
) (√
) ( *
Mathematical Analysis of Great Rhombicosidodecahedron/The Largest Archimedean Solid
Applications of “HCR’s Theory of Polygon” proposed by Mr H.C. Rajpoot (year-2014) ©All rights reserved
Hence, by substituting the corresponding value of in the above expression, we get
(
√( √ √ )
( √ √ )
)
(√ √
( √ ))
(√ √
( √ )) (√
( √ )( √ )
( √ )( √ ))
(√ √
) (
√ √
)
(
√ √
)
Normal distance ( ) of regular decagonal faces from the centre of great
rhombicosidodecahedron: The normal distance ( ) of each of 12 congruent regular decagonal faces from
the centre O of a great rhombicosidodecahedron is given from eq(V) as follows
√
( √ )
√ (
√ √ )
( √ )
√ √ ( √ )
√ √
√ √
It’s clear that all 12 congruent regular decagonal faces are at an equal normal distance from the centre
of any great rhombicosidodecahedron.
Solid angle ( ) subtended by each of the regular decagonal faces at the centre of great
rhombicosidodecahedron: solid angle ( ) subtended by each regular decagonal face is given from eq(VI)
as follows
(
√
( √
)
)
(
√
( √
)
)
(
*
Hence, by substituting the corresponding value of in the above expression, we get
(
√( √
) ( √ √ )
( √ √ )
)
(√( √ )( √ )
( √ ))
Mathematical Analysis of Great Rhombicosidodecahedron/The Largest Archimedean Solid
Applications of “HCR’s Theory of Polygon” proposed by Mr H.C. Rajpoot (year-2014) ©All rights reserved
(√ √
( √ )) (√
( √ )( √ )
( √ )( √ ))
(√ √
( )) (√
√
) (√
√
)
(√( √ )
, (
√
√ )
( √
√ )
It’s clear from the above results that the solid angle subtended by each of 12 regular decagonal faces is greater
than the solid angle subtended by each of 30 square faces & each of 20 regular hexagonal faces at the centre
of any great rhombicosidodecahedron.
It’s also clear from the above results that i.e. the normal distance ( ) of square faces is
greater than the normal distance of the regular hexagonal faces & the normal distance of the regular
decagonal faces from the centre of a great rhombicosidodecahedron i.e. regular decagonal faces are closer to
the centre as compared to the square & regular hexagonal faces in any great rhombicosidodecahedron.
Important parameters of a great rhombicosidodecahedron:
1. Inner (inscribed) radius ( ): It is the radius of the largest sphere inscribed (trapped inside) by a
great rhombicosidodecahedron. The largest inscribed sphere always touches all 12 congruent regular
decagonal faces but does not touch any of 30 congruent square & any of 20 congruent regular
hexagonal faces at all since all 12 decagonal faces are closest to the centre in all the faces. Thus, the
inner radius is always equal to the normal distance ( ) of regular decagonal faces from the centre of
a great rhombicosidodecahedron & is given as follows
√ √
Hence, the volume of inscribed sphere is given as
( )
(
√ √ )
2. Outer (circumscribed) radius ( ): It is the radius of the smallest sphere circumscribing a great
rhombicosidodecahedron or it’s the radius of a spherical surface passing through all 120 vertices of a
great rhombicosidodecahedron. It is from the eq(VII) as follows
√ √
Hence, the volume of circumscribed sphere is given as
( )
(
√ √ )
Mathematical Analysis of Great Rhombicosidodecahedron/The Largest Archimedean Solid
Applications of “HCR’s Theory of Polygon” proposed by Mr H.C. Rajpoot (year-2014) ©All rights reserved
3. Surface area ( ): We know that a great rhombicosidodecahedron has 30 congruent square faces,
20 congruent regular hexagonal faces & 12 congruent regular decagonal faces each of edge length .
Hence, its surface area is given as follows
( ) ( ) ( )
We know that area of any regular n-polygon with each side of length is given as
Hence, by substituting all the corresponding values in the above expression, we get
(
* (
* (
*
√ √ √ ( √ √ √ )
( √ √ √ )
4. Volume ( ): We know that a great rhombicosidodecahedron with edge length has 30 congruent
square faces, 20 congruent regular hexagonal faces & 12 congruent regular decagonal faces. Hence,
the volume (V) of the great rhombicosidodecahedron is the sum of volumes of all its elementary right
pyramids with square base, regular hexagonal base & regular decagonal base (face) (see figure 2
above) & is given as follows
( )
( )
( )
(
( ) * (
( ) *
(
( ) *
(
(
*
( √ )
) (
(
*
√ ( √ )
)
(
(
*
√ √ )
( √ ) ( √ ) √( √ )√( √ )
( √ ) √( √ ) ( √ ) ( √ ) ( √ )
( √ )
5. Mean radius ( ): It is the radius of the sphere having a volume equal to that of a great
rhombicosidodecahedron. It is calculated as follows
Mathematical Analysis of Great Rhombicosidodecahedron/The Largest Archimedean Solid
Applications of “HCR’s Theory of Polygon” proposed by Mr H.C. Rajpoot (year-2014) ©All rights reserved
( )
( √ ) ⇒ ( )
( √ )
(
( √ )
)
( ( √ )
)
It’s clear from above results that
6. Dihedral angles between the adjacent faces: In order to calculate dihedral angles between the
different adjacent faces with a common edge in a great rhombicosidodecahedron, let’s consider one-
by-one all three pairs of adjacent faces with a common edge as follows
a. Angle between square face & regular hexagonal face: Draw the
perpendiculars from the centre O of great
rhombicosidodecahedron to the square face & the regular hexagonal face
which have a common edge (See figure 3). We know that the inscribed radius
( ) of any regular n-gon with each side is given as follows
√
In right
( )
(( √ )
)
( )
( √ )
( √ ) ( )
In right
( √ )
(√ ( √ )
)
( √ )
( √ )
( √ ) ( )
⇒ ( √ ) ( √ ) (( √ ) ( √ )
( √ )( √ ))
( √
( √ )) (
( √ )
√ ) (
√
√ )
Figure 3: Square face with centre 𝑶𝟏 & regular hexagonal face with centre 𝑶𝟐 with a common edge (denoted by point T) normal to the plane of paper
Mathematical Analysis of Great Rhombicosidodecahedron/The Largest Archimedean Solid
Applications of “HCR’s Theory of Polygon” proposed by Mr H.C. Rajpoot (year-2014) ©All rights reserved
(( √ )( √ )
( √ )( √ )) (
√
) (
√
)
Hence, dihedral angle between the square face & the regular hexagonal face is given as
( √
)
b. Angle between square face & regular decagonal face: Draw the perpendiculars from the
centre O of great rhombicosidodecahedron to the square face & the regular decagonal face which
have a common edge (See figure 4).
√ √
In right
( √ √
+
( √ √ )
( √ √
+
√ ( √ )
√ √
(√ ) ( )
⇒ ( √ ) (√ ) (( √ ) (√ )
( √ )(√ ))
(( √ )
( √ )) (
( √ )
√ ) (
√
√ )
(( √ )( √ )
( √ )( √ )) (
(√ )
) (
√
)
Hence, dihedral angle between the square face & the regular decagonal face is given as
(√
)
c. Angle between regular hexagonal face & regular decagonal face: Draw the perpendiculars
from the centre O of great rhombicosidodecahedron to the regular hexagonal face & the
regular decagonal face which have a common edge (See figure 5). Now from eq(IX) & (X), we get
( √ ) (√ ) (( √ ) (√ )
( √ )(√ )) (
( √ )
( √ ))
( ( √ )
√ ) (
√
√ )
Figure 4: Square face with centre 𝑶𝟏 & regular decagonal face with centre 𝑶𝟑 with a common edge (denoted by point U) normal to the plane of paper
Mathematical Analysis of Great Rhombicosidodecahedron/The Largest Archimedean Solid
Applications of “HCR’s Theory of Polygon” proposed by Mr H.C. Rajpoot (year-2014) ©All rights reserved
(( √ )(√ )
(√ )(√ )) (
√
) ( √ )
Hence, dihedral angle between the regular hexagonal face & the regular
decagonal face is given as
( √ )
Construction of a solid great rhombicosidodecahedron: In order to
construct a solid great rhombicosidodecahedron with edge length there are two methods
1. Construction from elementary right pyramids: In this method, first we construct all elementary right
pyramids as follows
Construct 30 congruent right pyramids with square base of side length & normal height ( )
( √ )
Construct 20 congruent right pyramids with regular hexagonal base of side length & normal height ( )
√ ( √ )
Construct 12 congruent right pyramids with regular decagonal base of side length & normal height ( )
√ √
Now, paste/bond by joining all these elementary right pyramids by overlapping their lateral surfaces & keeping
their apex points coincident with each other such that 4 edges of each square base (face) coincide with the
edges of 2 regular hexagonal bases & 2 regular decagonal bases (faces). Thus a solid great
rhombicosidodecahedron, with 30 congruent square faces, 20 congruent regular hexagonal faces & 12
congruent regular decagonal faces each of edge length , is obtained.
2. Facing a solid sphere: It is a method of facing, first we select a blank as a solid sphere of certain material
(i.e. metal, alloy, composite material etc.) & with suitable diameter in order to obtain the maximum desired
edge length of a great rhombicosidodecahedron. Then, we perform the facing operations on the solid sphere
to generate 30 congruent square faces, 20 congruent regular hexagonal faces & 12 congruent regular
decagonal faces each of equal edge length.
Let there be a blank as a solid sphere with a diameter D. Then the edge length , of a great
rhombicosidodecahedron of the maximum volume to be produced, can be co-related with the diameter D by
relation of outer radius ( ) with edge length ( ) of the great rhombicosidodecahedron as follows
√ √
Figure 5: Regular hexagonal face with centre 𝑶𝟐 & regular decagonal face with centre 𝑶𝟑 with a common edge (denoted by point V) normal to the plane of paper
Mathematical Analysis of Great Rhombicosidodecahedron/The Largest Archimedean Solid
Applications of “HCR’s Theory of Polygon” proposed by Mr H.C. Rajpoot (year-2014) ©All rights reserved
Now, substituting ⁄ in the above expression, we have
√ √
√ √
√ √
Above relation is very useful for determining the edge length of a great rhombicosidodecahedron to be
produced from a solid sphere with known diameter D for manufacturing purpose.
Hence, the maximum volume of great rhombicosidodecahedron produced from a solid sphere is given as
follows
( √ ) ( √ ) (
√ √ )
( √ )
( √ )√ √
( √ )( √ )
√ √ ( √ )
√ √ ( √ )
√ √
( √ )
√ √
Minimum volume of material removed is given as
( ) ( )
( )
( √ )
√ √ (
( √ )
√ √ )
( ) (
( √ )
√ √ )
Percentage ( ) of minimum volume of material removed
(
( √ )
√ √ )
( ( √ )
√ √ )
It’s obvious that when a great rhombicosidodecahedron of the maximum volume is produced from a solid
sphere then about volume of material is removed as scraps. Thus, we can select optimum diameter
of blank as a solid sphere to produce a solid great rhombicosidodecahedron of the maximum volume (or with
the maximum desired edge length)
Conclusions: Let there be any great rhombicosidodecahedron having 30 congruent square faces, 20
congruent regular hexagonal faces & 12 congruent regular decagonal faces each with edge length then
all its important parameters are calculated/determined as tabulated below
Mathematical Analysis of Great Rhombicosidodecahedron/The Largest Archimedean Solid
Applications of “HCR’s Theory of Polygon” proposed by Mr H.C. Rajpoot (year-2014) ©All rights reserved
Congruent polygonal faces
No. of faces
Normal distance of each face from the centre of the great rhombicosidodecahedron
Solid angle subtended by each face at the centre of the great rhombicosidodecahedron
Square
30
( √ )
( √
)
Regular hexagon
20
√ ( √ )
(
√ √
)
Regular decagon
12
√ √
( √
√ )
Inner (inscribed) radius ( )
√ √
Outer (circumscribed) radius ( )
√ √
Mean radius ( )
( ( √ )
)
Surface area ( )
( √ √ √ )
Volume ( )
( √ )
Table for the dihedral angles between the adjacent faces of a great rhombicosidodecahedron
Pair of the adjacent faces with a common edge
Square & regular hexagon
Square & regular decagon
Regular hexagon & regular decagon
Dihedral angle of the corresponding pair (of the adjacent faces)
( √
)
(√
)
( √ )
Note: Above articles had been developed & illustrated by Mr H.C. Rajpoot (B Tech, Mechanical Engineering)
Mathematical Analysis of Great Rhombicosidodecahedron/The Largest Archimedean Solid
Applications of “HCR’s Theory of Polygon” proposed by Mr H.C. Rajpoot (year-2014) ©All rights reserved
M.M.M. University of Technology, Gorakhpur-273010 (UP) India March, 2015
Email: [email protected]
Author’s Home Page: https://notionpress.com/author/HarishChandraRajpoot